Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1

Limits and Derivatives Exercise 13.1 Solutions
1. Evaluate the Given limit (x + 3)
Solution
(x + 3) = 3 + 3 = 6
2. Evaluate the Given limit : (x - 22/7)
Solution
3. Evaluate the Given limit :
Ï€r2.
Solution
4. Evaluate the Given limit :
(4x + 3)/(x - 2)
Solution
5. Evaluate the Given limit :
(x10 + x5 + 1)/(x – 1)
Solution
6. Evaluate the Given limit :
[(x + 1)5 - 1]/x
Solution
7. Evaluate the Given limit :
(3x2 - x - 10)/(x2 - 4)
Solution
At x = 2, the value of the given rational function takes the form 0/0 .
8. Evaluate the Given limit
(x4 -81)/(2x2 - 5x - 3)
Solution
At x = 2, the value of the given rational function takes the form 0/0 .
9. Evaluate the Given limit :
(ax + b)/(cx + 1)
Solution
10. Evaluate the Given limit : 
Solution
At z = 1, the value of the given function takes the form 0/0 .
Put z1/6 = x so that z →1 as x →1.
11. Evaluate the Given limit
(ax2 + bx + c)/(cx2 + bx + a) , a + b + c ≠ 0 .
Solution
= 1 [a + b + c ≠ 0 ]
12. Evaluate the Given limit : 
Solution
At x = -2, the value of the given function takes the form 0/0 .
13. Evaluate the Given limit
sin ax/bx
Solution
At x = 0, the value of the given function takes the form 0/0 .
14. Evaluate the Given limit :
sin ax/sin bx, a,b ≠ 0
Solution
At x = 0, the value of the given function takes the form 0/0.
15. Evaluate the Given limit
[sin(Ï€ -x)]/Ï€(Ï€ - x)
Solution
16. Evaluate the given limit
cos x/(Ï€ - x)
Solution
17. Evaluate the Given limit :
(cos 2x - 1)/(cos x- 1)
Solution
At x = 0, the value of the given function takes the form 0/0 .
Now,
18. Evaluate the Given limit
(ax + x cos x)/b sin x
Solution
At x = 0, the value of the given function takes the form 0/0.
Now,
19. Evaluate the Given limit
sec x
Solution
20. Evaluate the Given limit
(sin ax + bx)/(ax + sin bx) a,b, a+b ≠ 0.
Solution
At x = 0, the value of the given function takes the form 0/0 .
Now,

Now,
21. Evaluate the Given limit
(cosec x - cot x)
Solution
At x = 0, the value of the given function takes the form ∞→∞.
Now,
Now,
22. Evaluate the Given limit
Solution
At x = π/2 , the value of the given function takes the form 0/0.
Now, put x - Ï€/2 = y so that x → Ï€/2, y → 0.
Solution
The given function is 

24. Find
f(x) where 
Solution
The given function is

25. Evaluate
f(x) where 
Solution
It is observed that
Hence,
26. Find
f(x) where 
Solution
The given function is

It is observed that
f(x) ≠
f(x).
Hence,
f(x) does not exist .
It is observed that
Hence,
27. Find
f(x) , where f(x) = |x| - 5
Solution
The given functions is f(x) = |x| - 5.

28. Suppose
and if
f(x) = f(1) what are possible values of a and b?
Solution
The given function is

f(x) =
(a + bx) = a + b
f(x) =
(b - ax) = b - a
f(1) = 4
It is given that
f(x) = f(1).
∴
f(x) =
f(x) =
f(x) = f(1)
⇒ a + b = 4 and b - a = 4
On solving these two equations, we obtain a = 0 and b = 4.
Thus, the respective possible values of a and b are 0 and 4.
f(1) = 4
It is given that
∴
⇒ a + b = 4 and b - a = 4
On solving these two equations, we obtain a = 0 and b = 4.
Thus, the respective possible values of a and b are 0 and 4.
29. Let a1, a2, . . ., an be fixed real numbers and define a function f ( x) = (x − a1) (x − a2)...(x − an). What is
f(x) ? For some a ≠ a1, a2, ..., an, compute 
Solution
The given function is f(x) = (x – a1)(x – a2) … (x - an).

30. If
For what value (s) of a does
f(x) exists ?
Solution
The given function is

When a = 0,

When a = 0,
31. If the function f(x) satisfies
[f(x) - 2]/[x2 -1] = π , evaluate
f(x).
Solution
32. If 
For what integers m and n does
f(x) and
f(x) exist ?
For what integers m and n does
Solution
The given function is
